Signed Distance Fields (SDFs) parameterized by neural networks have recently gained popularity as a fundamental geometric representation. However, editing the shape encoded by a neural SDF remains an open challenge. A tempting approach is to leverage common geometric operators (e.g., boolean operations), but such edits often lead to incorrect non-SDF outputs (which we call Pseudo-SDFs), preventing them from being used for downstream tasks. In this paper, we characterize the space of Pseudo-SDFs, which are eikonal yet not true distance functions, and derive the closest point loss, a novel regularizer that encourages the output to be an exact SDF. We demonstrate the applicability of our regularization to many operations in which traditional methods cause a Pseudo-SDF to arise, such as CSG and swept volumes, and produce a true (neural) SDF for the result of these operations.

Geometry processing presents a variety of difficult numerical problems, each seeming to require its own tailored solution. This breadth is largely due to the expansive list of geometric primitives , e.g., splines, triangles, and hexahedra, joined with an ever-expanding variety of objectives one might want to achieve with them. With the recent increase in attention toward higher-order surfaces , we can expect a variety of challenges porting existing solutions that work on triangle meshes to work on these more complex geometry types. In this paper, we present a framework for solving many core geometry processing problems on higher-order surfaces. We achieve this goal through sum-of-squares optimization, which transforms nonlinear polynomial optimization problems into sequences of convex problems whose complexity is captured by a single degree parameter. This allows us to solve a suite of problems on higher-order surfaces, such as continuous collision detection and closest point queries on curved patches, with only minor changes between formulations and geometries.

Abstract The validity of trilinear hexahedral (hex) mesh elements is a prerequisite for many applications of hex meshes, such as finite element analysis. A commonly used check for hex mesh validity evaluates mesh quality on the corners of the parameter domain of each hex, an insufficient condition that neglects invalidity elsewhere in the element, but is straightforward to compute. Hex mesh quality optimizations using this validity criterion suffer by being unable to detect invalidities in a hex mesh reliably, let alone fix them. We rectify these challenges by leveraging sum‐of‐squares relaxations to pinpoint invalidities in a hex mesh efficiently and robustly. Furthermore, we design a hex mesh repair algorithm that can certify validity of the entire hex mesh. We demonstrate our hex mesh repair algorithm on a dataset of meshes that include hexes with both corner and face‐interior invalidities and demonstrate that where naïve algorithms would fail to even detect invalidities, we are able to repair them. Our novel methodology also introduces the general machinery of sum‐of‐squares relaxation to geometry processing, where it has the potential to solve related problems.